Functions: Definitions and Notation

1.3 – 1.4

P 43-75 (text)

Pages 55-87 (pdf)

Mapping DiagramA mapping diagram is a diagram that can illustrate the relationship between the domain and range of a relation.

It can only be used if the relation is finite or if a recognizable pattern exists between the domain

and range.Domain Range

-1

0

1

3

-3

3

4

5

Function – Definition and Notation

A function is a relation in which each element of the domain is paired with exactly one element of the range.

Symbolic Notation

f(x)

Any element of the domain

The name of the function is “f”

Any element of the range

Notation

The symbolic notation f(x) can be read as:

“f of x”

“f at x” or

“the value of the function f at x”

The most common method

Ways to Express Functions

Functions can be expressed in many ways:

Roster or List

Mapping Diagram

Equation or Rule

GraphWe will use these methods most often.

Function NotationAn equation or rule representing a function can also be expressed in several ways. Some examples are:

2f:x 3 2 Mapping notationx x 2f(x)= 3 2 Standard function notationx x

2 3 2 0 Implicit Form (xy-notation)x x y

2 3 2 Explicit Form (xy-notation)y x x

2(x, f(x)): f(x)= 3 2 Set notationx x

Expressing Functions Graphically

2f(x)= 3 2 x x

A function can be expressed graphically by plotting points in a coordinate plane.

–6 –5 –4 –3 –2 –1 1 2 3

–1

1

2

3

4

5

6

7

8

x

y

–6 –4 –2 2 4 6

–4

–3

–2

–1

1

2

3

4

x

y

f(x)=3sin(2x)

Determining Whether a Graph is a Function

In a function, since each x-value must be paired with exactly one y-value, no x-values can repeat. This gives way to the vertical line test.The Vertical Line Test states: A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.

Use of the Vertical Line TestDetermine whether each of the following graphs represents a function.

–2 –1 1 2 3 4 5 6

–12

–10

–8

–6

–4

–2

2

4

x

y

Graph is a function

–6 –4 –2 2 4 6

–6

–4

–2

2

4

6

x

y

Graph is NOT a function

Finding the Domain of a FunctionThe domain of a function is the set of all x-values for which the function is defined.

We can determine the domain of a function if we are given the equation of the function or the graph of the function.

–6 –4 –2 2 4 6

–6

–4

–2

2

4

6

x

y

The domain of the function is all real numbers. Notice that the graph extends infinitely upward to the left and right.

Determining the Domain of a Function

: 3 3x x

–6 –4 –2 2 4 6

–6

–4

–2

2

4

6

x

y

The domain of the function is all real values of x from -3 to 3. In set notation it is:

–6 –4 –2 2 4 6

–6

–4

–2

2

4

6

x

y

The domain of the function is all real numbers greater than or equal to -4. In set notation it is

: 4x x

Determining the Domain of a Function from the Equation.

2( ) 5f x x

1

1y

x

Regardless of the x-value chosen there will always be a corresponding value of f(x). Therefore, the domain is the set of real numbers.

Since the formula for the function contains a fraction, the value of x that makes the denominator zero cannot be chosen as a value of x. Therefore, the domain is all real numbers except 1. In set notation:

: , 1x x R x

Determining the Domain of a Function from the Equation.

For this function to be defined using the real number system, the radicand must be non-negative. So 2x + 4 > 0. Solving for x, x> -2. Therefore, the domain of the function is {x: xeR, x> -2}

–6 –4 –2 2 4 6

–2

–1

1

2

3

4

5

6

x

y

Domain

 

Finding values of f(x)

F(x) = 2x2- 8x + 8

What is the value of f(0)?

This is the x-intercept. Substitute 0 into the equation and solve

(0,8)

Exit Ticket:

1. Find f(4)

2. Find f(x+3)

_______________________________

Homework:

Text: 1.3 Q 33, 36, 39, 42, 45 (p 52)

1.4 Q 36, 40, 44, 48, 52 (p65)